I was able to hear Cliburn live several times during the early 1960's when he visited Munich. I have an autographed copy of his Rach PC 3 LP which he signed for me in 1962. I also met his mother, a delightful person.
The tragedy of Cliburn's career is that both his artistry and technique went into a rapid decline in the late sixties and never recovered. We all thought he'd be the next Rubinstein, and instead, he turned out to be the next Jackie du Pre (without the MS, of course).
The two concerto recordings made on his return from Moscow with Kondrashin are must-have monuments of the phonograph. His Tchai PC1, for all its popularity, remains the supreme recording of this work. The live Rach PC 3 from earlier in the same evening is also a benchmark, while adding the palpable tension and sheer excitement of the live event.
Other great concerto recordings include the McDowell PC2 and Prok 3 (with Hendl and the CSO), the Schumann with Reiner, and the later Brahms Bflat with Kondrashin. I would pay real money for an aircheck (any quality) of him performing the Schumann in Hollywood Bowl, 1959, conducted by Bruno Walter.
The Rach sonata (live from Moscow), "My Favorite Chopin", the Debussy and Brahms discs, all are stellar.